Since the group \(\mathrm {SU}(2)\)  is compact the Nelson Laplacian is the quadratic Casimir operator, see for instance [1]}.
By irreducibility (due to Schur's Lemma), it is a multiple of the identity, namely \(\Delta \equiv \vec{S}^2 =j(j+1)\text{\usefont {U}{bbold}{m}{n}1}\) .
Therefore using the Nelson Laplacian, thm:main gives the bound
\({U^{(j)}_g - U^{(j)}_h} \le j(j+1) \, {\log (g^{-1}h)}_2.\) 
